We show that the spectrum of the Dirac operator in complex Langevin simulations of QCD at non-zero chemical potential must behave in a way which is radically different from the one in simulations with ordinary non-complexified gauge fields: At low temperatures the small eigenvalues of the Dirac operator must be inside the quark mass for chemical potentials as large as a third of the nucleon mass. In particular, in the chiral limit the Dirac eigenvalues of complex Langevin simulations must accumulate at the origin.
We employ the Complex Langevin method for simulation of complex-valued actions. First, we show how to test for convergence of the method by explicitely computing boundary terms and demonstrate this in a model. Then we investigate the deconfinement phase transition of QCD with $N_f=2$ Wilson-fermions using the Complex Langevin Method and. We give preliminary results for the transition temperatures up to $μ/T_c(μ=0)\approx 5$ and compute the curvature coefficient $κ_2$.
The sign problem of QCD prevents standard lattice simulations to determine the phase diagram of strong interactions with a finite chemical potential directly. Complex Langevin simulations provide an alternative method to sample path integrals with complex weights. We report on our ongoing project to determine the phase diagram of QCD in the limit of heavy quarks (HDQCD) using Complex Langevin simulations.
We present the reweighted complex Langevin method, which enlarges the applicability range of the complex Langevin method by reweighting the complex trajectories. In this reweighting procedure both the auxiliary and target ensembles have a complex action. We validate the method by applying it to two-dimensional strong-coupling QCD at nonzero chemical potential, and observe that it gives access to parameter regions that could otherwise not be reached with the complex Langevin method.
Monte Carlo studies of QCD at finite density suffer from the sign problem, which becomes easily uncontrollable as the chemical potential $μ$ is increased even for a moderate lattice size. In this work we make an attempt to approach the high density low temperature region by the complex Langevin method (CLM) using four-flavor staggered fermions with reasonably small quark mass on a $8^3 \times 16$ lattice. Unlike the previous work on a $4^3 \times 8$ lattice, the criterion for correct convergence is satisfied within a wide range of $μ$ without using the deformation technique. In particular, the…
Progress in the application of the complex Langevin method to full QCD at non-zero chemical potential is reported. The method evades the sign problem which makes naive simulations at nonzero density impossible. The procedure 'gauge cooling' is used to stabilize the simulations at small enough lattice spacings. The method allows simulations also at high densities, all the way up to saturation. Simulations in a systematic hopping parameter expansion are also performed and good convergence is observed, validating the full as well as the expanded simulations.
In this review we present the current state-of-the-art on complex Langevin simulations and their implications for the QCD phase diagram. After a short summary of the complex Langevin method, we present and discuss recent developments. Here we focus on the explicit computation of boundary terms, which provide an observable that can be used to check one of the criteria of correctness explicitly. We also present the method of Dynamic Stabilization and elaborate on recent results for fully dynamical QCD.
We study full QCD at finite density and low temperature with light quark mass using the complex Langevin method. Since the singular drift problem turns out to be mild on a $4^3 \times 8$ lattice we use, the gauge cooling is performed only to control the unitarity norm in this exploratory study. We report on our preliminary data obtained from the complex Langevin simulation up to certain Langevin time. While the data are still noisy due to lack of statistics, the onset of the baryon number density seems to occur at larger $μ$ than half the pion mass, which is the value for the phase quenched QC…
Complex Langevin simulations provide an alternative to sample path integrals with complex weights and therefore are suited to determine the phase diagram of QCD from first principles. We use our proposed method of Dynamic Stabilisation (DS) to ensure improved convergence to the right limit and present new systematic tests of this technique. We also show results on QCD in the limit of heavy quarks and an analysis of DS compared to known results from reweighting.
Lattice simulations of non-zero density QCD introduce the so-called sign problem (complex or negative probabilities), which invalidates importance sampling methods. To circumvent this, we use the Complex Langevin Equation (CLE), to measure the boundary terms and then compare these results with the ones gotten from reweighting, confirming the expectations from previous studies. We also investigate boundary terms in simulations using CLE with dynamic stabilization and compare this, to results calculated with reweighting.
We present our results obtained from gauge cooled complex Langevin simulations in 1+1d QCD at non-zero densities in the strong coupling regime with unrooted staggered fermions. For small quark masses there are regions of the chemical potential where this method fails to reproduce correct results. In these parameter ranges we studied the effect of different gauge cooling schemes on the distributions of the fermion determinant as well as of observables.
Monte Carlo methods cannot probe far into the QCD phase diagram with a real chemical potential, due to the famous sign problem. Complex Langevin simulations, using adaptive step-size scaling and gauge cooling, are suited for sampling path integrals with complex weights. We report here on tests of the deconfinement transition in pure Yang-Mills SU(3) simulations and present an update on the QCD phase diagram in the limit of heavy and dense quarks.
We present our latest results on the application of the complex Langevin method to one- and two-dimensional QCD. Although the method is stable, it unfortunately converges to an incorrect result when applied as such. After applying additional gauge cooling steps, the results agree with the known analytical results in the one-dimensional case. However, in the two-dimensional case the disagreement subsists, even with gauge cooling, when the sign problem is sufficiently large.
Complex Langevin simulations provide an alternative to sample path integrals with complex weights and therefore are suited to determine the phase diagram of QCD from first principles. Adaptive step-size scaling and gauge cooling are used to improve the convergence of our simulations. We present results for the phase diagram of QCD in the limit of heavy quarks and discuss the order of the phase transitions, which are studied by varying the spatial simulation volume.
We apply the complex Langevin (CL) method to a chiral random matrix theory (ChRMT) at non-zero chemical potential and study the nearest neighbor spacing (NNS) distribution of the Dirac eigenvalues. The NNS distribution is extracted using an unfolding procedure for the Dirac eigenvalues obtained in the CL method. For large quark mass, we find that the NNS distribution obeys the Ginibre ensemble as expected. For small quark mass, the NNS distribution follows the Wigner surmise for correct convergence case, while it follows the Ginibre ensemble for wrong convergence case. The Wigner surmise is ph…
Simulations of QCD with a finite chemical potential typically lead to a severe sign problem, prohibiting any standard Monte Carlo approach. Complex Langevin simulations provide an alternative to sample path integrals with oscillating weight factors and therefore potentially enable the determination of the phase diagram of QCD. Here we present results for QCD in the limit of heavy quarks and show evidence that the phase diagram can be mapped out by direct simulation. We apply adaptive step-size scaling and adaptive gauge cooling to ensure the convergence of these simulations.
QCD at nonzero baryon chemical potential suffers from the sign problem, due to the complex quark determinant. Complex Langevin dynamics can provide a solution, provided certain conditions are met. One of these conditions, holomorphicity of the Langevin drift, is absent in QCD since zeroes of the determinant result in a meromorphic drift. We first derive how poles in the drift affect the formal justification of the approach and then explore the various possibilities in simple models. The lessons from these are subsequently applied to both heavy dense QCD and full QCD, and we find that the resul…
Complex Langevin simulations have been able to successfully reproduce results from Monte Carlo methods in the region where the sign problem is mild and make predictions when it is exponentially hard. We present here our study of the QCD phase diagram and the boundary between the confined and deconfined phases in the limit of heavy and dense quarks (HDQCD) for 3 different lattice volumes. We also briefly discuss instabilities encountered in our simulations.
We present lattice simulations on the phase diagram of Quantum Chromodynamics (QCD) with two light quark flavours at finite chemical potential $μ$. To circumvent the sign problem we use the complex Langevin method. In this study, we have carried out finite density lattice computations for a pion mass of $\sim 480$ MeV. We report on the pressure, energy and entropy equations of state in ab-initio lattice QCD calculations, as well as the observation of the Silver Blaze phenomenon.
The ability to describe strongly interacting matter at finite temperature and baryon density provides the means to determine, for instance, the equation of state of QCD at non-zero baryon chemical potential. From a theoretical point of view, direct lattice simulations are hindered by the numerical sign problem, which prevents the use of traditional methods based on importance sampling. Despite recent successes, simulations using the complex Langevin method have been shown to exhibit instabilities, which cause convergence to wrong results. We introduce and discuss the method of Dynamic Stabilis…
Complex Langevin simulations allow numerical studies of theories that exhibit a sign problem, such as QCD, and are thereby potentially suitable to determine the QCD phase diagram from first principles. Here we study QCD in the limit of heavy quarks for a wide range of temperatures and chemical potentials. Our results include an analysis of the adaptive gauge cooling technique, which prevents large excursions into the non-compact directions of the SL($3, \mathbb{C}$) manifold. We find that such excursions may appear spontaneously and change the statistical distribution of physical observables, …
We use complex Langevin simulations to explore the QCD phase diagram over a large range of chemical potentials and temperatures. For our simulations, we use two flavours of dynamical Wilson fermions with a pion mass of approximately $480\,$MeV with a spatial volume of $24^3$. Here we report on consistency checks at zero chemical potentials and present our results for the fermion density and the Polyakov loop. We find that at the lowest temperature the fermion density remains zero until $m_N/3$, in line with the expectations from the Silver Blaze phenomenon.
Recently, the complex Langevin method has been applied successfully to finite density QCD either in the deconfinement phase or in the heavy dense limit with the aid of a new technique called the gauge cooling. In the confinement phase with light quarks, however, convergence to wrong limits occurs due to the singularity in the drift term caused by small eigenvalues of the Dirac operator including the mass term. We propose that this singular-drift problem should also be overcome by the gauge cooling with different criteria for choosing the complexified gauge transformation. The idea is tested in…
We study the Stephanov model, which is an RMT model for QCD at finite density, using the Complex Langevin algorithm. Naive implementation of the algorithm shows convergence towards the phase quenched or quenched theory rather than to intended theory with dynamical quarks. A detailed analysis of this issue and a potential resolution of the failure of this algorithm are discussed. We study the effect of gauge cooling on the Dirac eigenvalue distribution and time evolution of the norm for various cooling norms, which were specifically designed to remove the pathologies of the complex Langevin evo…
Exploring the phase diagram of QCD at finite density is a challenging problem since first-principle calculations based on standard Monte Carlo methods suffer from the sign problem. As a promising approach to this issue, the complex Langevin method (CLM) has been pursued intensively.In this work, we investigate the applicability of the CLM in the vicinity of the deconfinement phase transition using the four-flavor staggered fermions. In particular, we look for a signal of the expected first order phase transition within the validity region of the CLM.
We study lattice QCD at non-vanishing chemical potential using the complex Langevin equation. We compare the results with multi-parameter reweighting both from $μ=0$ and phase quenched ensembles. We find a good agreement for lattice spacings below $\approx$0.15 fm. On coarser lattices the complex Langevin approach breaks down. Four flavors of staggered fermions are used on $N_t=4, 6$ and 8 lattices. For one ensemble we also use two flavors to investigate the effects of rooting.
We study a random matrix model for QCD at finite density via complex Langevin dynamics. This model has a phase transition to a phase with nonzero baryon density. We study the convergence of the algorithm as a function of the quark mass and the chemical potential and focus on two main observables: the baryon density and the chiral condensate. For simulations close to the chiral limit, the algorithm has wrong convergence properties when the quark mass is in the spectral domain of the Dirac operator. A possible solution of this problem is discussed.
We propose two novel formulations of the hopping parameter expansion for finite density QCD using Wilson fermions, while keeping the gauge action intact. We use the complex Langevin equation to circumvent the sign problem in the theory. We perform simulations at very high order of the expansion, such that convergence is directly observable. We compare results to the full QCD results, and see agreement at sufficiently high orders. These results provide support for the use of complex Langevin dynamics to study QCD at nonzero density, both in the full and the expanded theory, and for the converge…
We present new results of full QCD at nonzero chemical potential. In PRD 92, 094516 (2015) the complex Langevin method was shown to break down when the inverse coupling decreases and enters the transition region from the deconfined to the confined phase. We found that the stochastic technique used to estimate the drift term can be very unstable for indefinite matrices. This may be avoided by using the full inverse of the Dirac operator, which is, however, too costly for four-dimensional lattices. The major breakthrough in this work was achieved by realizing that the inverse elements necessary …
We discuss the flavor number dependence of QCD at low temperature and high density by the complex Langevin method. In our previous work, the complex Langevin method is confirmed to satisfy the criterion for correct convergence in certain regions, such as $μ_{\rm q} / T = 5.2-7.2$ on $8^3 \times 16$ and $μ_{\rm q} / T = 1.6-9.6$ on $16^3 \times 32$ using $N_{\rm f} = 4$ staggered fermion at $β= 5.7$. We extend this study to more realistic flavor cases, $N_{\rm f} = 2, 2 + 1, 3$, using Wilson fermions. We present the flavor number dependence of the validity regions of the complex Langevin method…
QCD at non-zero chemical potential ($μ$) for quark number has a complex fermion determinant and thus standard simulation methods for lattice QCD cannot be applied. We therefore simulate this theory using the Complex-Langevin algorithm with Gauge Cooling in addition to adaptive methods, to prevent runaway behaviour. Simulations are performed at zero temperature on a $12^4$ lattice with 2 quarks which are light enough that $m_N/3$ is significantly larger than $m_π/2$. Preliminary results are qualitatively as expected. The quark-number density is close to zero for $μ< m_N/3$, beyond which it incr…
Complex Langevin methods have been successfully applied in theories that suffer from a sign problem such as QCD with a chemical potential. We present and illustrate a novel method (dynamic stabilisation) that ensures that Complex Langevin simulations stay close to the SU(3) manifold, which lead to correct and improved results in the framework of pure Yang-Mills simulations and QCD in the limit of heavy quarks.
We study QCD at finite density and low temperature by using the complex Langevin method. We employ the gauge cooling to control the unitarity norm and introduce a deformation parameter in the Dirac operator to avoid the singular-drift problem. The reliability of the obtained results are judged by the probability distribution of the magnitude of the drift term. By making extrapolations with respect to the deformation parameter using only the reliable results, we obtain results for the original system. We perform simulations on a $4^3\times 3$ lattice and show that our method works well even in …
I review the status of the Complex Langevin method, which was invented to make simulations of models with complex action feasible. I discuss the mathematical justification of the procedure, as well as its limitations and open questions. Various pragmatic measures for dealing with the existing problems are described. Finally I report on the progress in the application of the method to QCD, with the goal of determining the phase diagram of QCD as a function of temperature and baryonic chemical potential.
The sign problem appears in lattice QCD as soon as a non-zero chemical potential is introduced. This prevents direct simulations to determine the phase structure of the strongly interacting matter. Complex Langevin methods have been successfully used for various models or approximations of QCD. However, in some scenarios it converges to incorrect results. We present developments of our new method that helps to improve the convergence by keeping the system closer to the SU(3) manifold and discuss preliminary tests and results.
Lattice QCD at non-vanishing chemical potential is studied using the complex Langevin equation (CLE). One of the conditions for the correctness of the results of the CLE is that the zeroes of the measure coming from the fermionic determinant are outside of the distribution of the configurations, or at least in a region where support for the distribution is very much suppressed. We investigate this issue for Heavy Dense QCD (HDQCD) and full QCD at high temperatures. In HDQCD it is found that the configurations move closest to the zeroes of the measure around the critical chemical potential of t…
We simulate lattice QCD at finite quark-number chemical potential to study nuclear matter, using the complex Langevin equation (CLE). The CLE is used because the fermion determinant is complex so that standard methods relying on importance sampling fail. Adaptive methods and gauge-cooling are used to prevent runaway solutions. Even then, the CLE is not guaranteed to give correct results. We are therefore performing extensive testing to determine under what, if any, conditions we can achieve reliable results. Our earlier simulations at $β=6/g^2=5.6$, $m=0.025$ on a $12^4$ lattice reproduced the…
It is well known that investigating QCD at finite density by standard Monte Carlo methods is extremely difficult due to the sign problem. Some years ago, the complex Langevin method with gauge cooling was shown to work at high temperature, i.e., in the deconfined phase. The same method was also applied to QCD in the so-called heavy dense limit in the whole temperature region. In this paper we attempt to apply this method to the large $μ/T$ regime with moderate quark mass using four-flavor staggered fermions on a $4^3\times 8$ lattice. While a straightforward application faces with the singular…
In application of the complex Langevin method to QCD at high density and low temperature, the singular-drift problem occurs due to the appearance of near-zero eigenvalues of the Dirac operator. In order to avoid this problem, we proposed to deform the Dirac operator in such a way that the near-zero eigenvalues do not appear and to extrapolate the deformation parameter to zero from the available data points. Here we test three different types of deformation in a simple large-$N$ matrix model, which undergoes an SSB due to the phase of the fermion determinant, and compare them to see the consist…
We explore the QCD phase diagram at finite density with four-flavor staggered fermions using the complex Langevin method, which is a promising approach to overcome the sign problem. In our previous work on an $8^3 \times 16$ lattice at $β= 5.7$ with the quark mass $m = 0.01$, we have found that the baryon number density has a clear plateau as a function of the chemical potential. In this study, we use a $16^3 \times 32$ lattice to reduce finite volume effects and find that the plateau structure survives. Moreover, the number of quarks in the plateau region turns out to be 24, which is exactly …
We study the use of the complex-Langevin equation (CLE) to simulate lattice QCD at a finite chemical potential ($μ$) for quark-number, which has a complex fermion determinant that prevents the use of standard simulation methods based on importance sampling. Recent enhancements to the CLE specific to lattice QCD inhibit runaway solutions which had foiled earlier attempts to use it for such simulations. However, it is not guaranteed to produce correct results. Our goal is to determine under what conditions the CLE yields correct values for the observables of interest. Zero temperature simulation…
Recent progress of the complex Langevin method and the Lefschetz thimble in connection with the sign problem is reviewed. These methods rely on the complexification of the original field manifold and they allow direct simulations of theories with non-real measures. Similarities and differences of the two approaches are pointed out. Results using the complex Langevin method, which allows simulations to evade the sign problem in full QCD, are presented. Promising results of the thimble approach for non-gauge theories are also discussed.
We derive exact analytical expressions for the spectral density of the Dirac operator at fixed θ-angle in the microscopic domain of one-flavor QCD. These results are obtained by performing the sum over topological sectors using novel identities involving sums of products of Bessel functions. Because the fermion determinant is not positive definite for negative quark mass, the usual Banks-Casher relation is not valid and has to be replaced by a different mechanism first observed for QCD at nonzero chemical potential. Using the exact results for the spectral density we explain how this mechanism…
We continue our simulations of lattice QCD at finite quark-number chemical potential, $μ$, using the complex-Langevin equation (CLE) with gauge-cooling and adaptive updating. The CLE is used because QCD at finite finite $μ$ has a complex fermion determinant, which prevents use of standard simulation methods. Simulations using the standard lattice action show a transition from hadronic to nuclear matter for $μ< m_π/2$ rather than the expected $μ\approx m_N/3$. This suggests that the CLE is being influenced by the phase-quenched theory, which has a transition at $μ= m_π/2$. We are therefore perf…
QCD at finite quark-/baryon-number density, which describes nuclear matter, has a sign problem which prevents direct application of standard simulation methods based on importance sampling. When such finite density is implemented by the introduction of a quark-number chemical potential $μ$, this manifests itself as a complex fermion determinant. We apply simulations using the Complex Langevin Equation (CLE) which can be applied in such cases. However, this is not guaranteed to give correct results, so that extensive tests are required. In addition, gauge cooling is required to prevent runaway …
We simulate lattice QCD at finite quark-number chemical potential, $μ$, using the complex-Langevin equation (CLE) with gauge-cooling and adaptive updating to prevent instabilities. The CLE is used because QCD at finite $μ$ has a complex fermion determinant which precludes the use of standard simulation methods based on importance sampling. Since, even when CLE simulations converge, they are not guaranteed to produce correct results except under very stringent conditions, which lattice QCD at finite $μ$ does not obey, we need extensive testing to determine under what conditions it produces reli…
It is known that contrary to expectations, the order parameter of chiral symmetry breaking, the Dirac spectral density at zero virtuality does not vanish above the critical temperature of QCD. Instead, the spectral density develops a pronounced peak at zero. We show that the spectral density in the peak has large violations of the expected volume scaling. This anomalous scaling and the statistics of these eigenmodes is consistent with them being produced by mixing instanton and antiinstanton zero modes. Consequently, we show that a nonvanishing topological usceptibility implies a finite densit…
